There are hundreds of places to go out to in Tokyo. It all depends what you want to do. Roppongi, 六本木, is the "going out" centre of Tokyo. Go there for hundreds of things to do. Nearest Underground: Roppongi.

I recommend.... Asakusa: Japanese old temple is there. Akihabara: Many funny shops are there. Shibuya, Harajuku: Many young people are there.　bustling, vibrant city. Roppongi: Many foreigners hang out and drink there. And you can sing at Karaoke shops, everywhere :) ( Same as chihiro_ito, I answered similar question before. http://www.italki.com/answers/question/69094.htm )

Tokyo is very modernized so if you wanna experience something traditional, I think the only place I can recommend you to go is Asakusa. As for the outside of Tokyo, Kamakura is nice.You can see a huge Budda statue and many temples. I guess it takes 40min from Tokyo sta to go to Kamakura, If you wanna enjoy today' s Japanese culture, Akihabara is good to buy manga related goods, and Shibuya/ Harajuku are good to see young Japanese fashions.

秋葉原 Akihabara and 巣鴨 Sugamo were interesting. =)

There are many. You will find them.